Pseudo-panelists
The logistics of collecting data that spans a subject’s lifetime through longitudinal research make it quite difficult to obtain very long-term data. Firstly, the data must necessarily be collected over a period of multiple decades (perhaps as many as 8 – 10). Secondly, the attrition rates for a multi-decade study are expected to be quite high. To compensate, an initial participant group several times larger than the desired final participant group size must be recruited and tracked. In addition, the costs associated with such an undertaking will be restrictive for all but the most financially leveragable areas of research. Finally, the results of the study will not be available for at least a lifetime, and repeated measures will require similar time and resources. It is difficult to envision a scenario whereby the return on investment is sufficiently high to justify the project, but the urgency is sufficiently low to allow decades for the study’s results to be revealed.
The present study investigates the relationship between juice consumption and BMI over the course of the subjects’ lifetimes. However, given the complications detailed above, a life-long longitudinal study cannot be justified. Instead, a relatively lesser-known technique of pseudo-panelists will be explored. The idea behind pseudo-panelists is that a comprehensive cross-sectional data set with many subjects can be reduced a data set equivalent to a longitudinal data set with fewer panelists. However, each “panelist” will be an aggregate of several subjects for each observation (age), hence the term “pseudo-panelist.”
The NHANES data with collection years beginning 1999 – 2009 feature 50,918 subjects over the 6 data sets collected over that time frame. Each individual data set featured a comparable number of subjects, with the six data sets demonstrating a standard deviation of 5%.
Table 1 – NHANES Data: Number of Subjects by Data Beginning Collection Year
Year Frequency Percent
1999 8335 16.4
2001 7788 15.3
2003 8483 16.7
2005 8797 17.3
2007 8486 16.7
2009 9029 17.7
Total 50918 100
The subjects in the NHANES data are aged 2 – 84 years. Childhood ages were more heavily represented. Adult ages from 20 – 50 years were fairly consistent in their representation. Participation dipped for ages 50 – 60, increased at age 60, and then declined from ages 60 – 84. See Figure 1, below.
Figure 1 – NHANES Data: Number of Subjects by Age
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“ARIMA(p,d,q): ARIMA models are, in theory, the most general class of models for forecasting a time series which can be stationarized by transformations such as differencing and logging. In fact, the easiest way to think of ARIMA models is as fine-tuned versions of random-walk and random-trend models: the fine-tuning consists of adding lags of the differenced series and/or lags of the forecast errors to the prediction equation, as needed to remove any last traces of autocorrelation from the forecast errors.
The acronym ARIMA stands for “Auto-Regressive Integrated Moving Average.” Lags of the differenced series appearing in the forecasting equation are called “auto-regressive” terms, lags of the forecast errors are called “moving average” terms, and a time series which needs to be differenced to be made stationary is said to be an “integrated” version of a stationary series. Random-walk and random-trend models, autoregressive models, and exponential smoothing models (i.e., exponential weighted moving averages) are all special cases of ARIMA models.
A nonseasonal ARIMA model is classified as an “ARIMA(p,d,q)” model, where:
• p is the number of autoregressive terms,
• d is the number of nonseasonal differences, and
• q is the number of lagged forecast errors in the prediction equation.
To identify the appropriate ARIMA model for a time series, you begin by identifying the order(s) of differencing needing to stationarize the series and remove the gross features of seasonality, perhaps in conjunction with a variance-stabilizing transformation such as logging or deflating. If you stop at this point and predict that the differenced series is constant, you have merely fitted a random walk or random trend model. (Recall that the random walk model predicts the first difference of the series to be constant, the seasonal random walk model predicts the seasonal difference to be constant, and the seasonal random trend model predicts the first difference of the seasonal difference to be constant–usually zero.) However, the best random walk or random trend model may still have autocorrelated errors, suggesting that additional factors of some kind are needed in the prediction equation.”
[From http://people.duke.edu/~rnau/411arim.htm]
Figure 12 – ARIMA Predictions for Pseudo-panel data
You find below a table illustrating the number of subjects who contributed to the data for each pseudo-panelist at each age.
Within the spreadsheet, on the “Fit” tab, you will find a number of plots demonstrating the fit of pseudo-panelist data to two ARIMA models. The first model, denoted “w/0s,” was created by computing the overall average juice consumption and BMI for all subjects at each binned age. These data were used to produce an ARIMA model. The resulting model was ARIMA(1,1,0) with the analytic form, BMI*(t) = BMI(t-1) + 0.825 * [BMI(t-1) – BMI(t-2)], where BMI*(t) is the forecast BMI for age, t, and BMI(t) if the observed BMI for age, t. The second model, denoted “wo/0s,” was created by computing the average juice consumption for all subjects who reported consuming juice (hence, without 0s) and the average BMI for the same subjects at each age. These data were best estimated by an ARIMA(0,2,0) model with the analytical form, BMI*(t) = -0.2 + 2 * BMI(t-1) – BMI(t-2). In either case, juice consumption was not a significant factor in predicting BMI.
The plots on this tab show the fit of the pseudo-panelist data to these aggregate models. In both cases the fits were fairly good.
MAPE for w/0s model by pseudo-panelist
0 1 2 3 4
2.96% 5.90% 4.76% 5.69% 3.85%
MAPE for wo/0s model by pseudo-panelist
0 1 2 3 4
2.35% 5.82% 4.73% 5.51% 3.41%
MAPE stands for Mean Absolute Percent Error, and the plots illustrate that the overall models do a reasonably good job of predicting BMI for the pseudo-panelists. Of course, since juice consumption was largely independent of BMI, the pseudo-panelists represent little more than random samples of the BMI data. From this perspective it is not surprising that the pseudo-panelists’ data were accurately forecast by the overall models.
The two models were also highly consistent with each other:
Mean average percent difference by pseudo-panelist
0 1 2 3 4
0.82% 0.93% 0.89% 0.95% 0.86%
Pseudo-panelist
Age 0 1 2 3 4
5 2233 502 502 502 503
10 3489 422 423 423 423
15 5050 469 470 470 470
20 4906 358 358 358 359
25 2195 149 149 149 150
30 2109 136 136 136 137
35 2016 133 134 133 134
40 2139 121 121 121 122
45 2147 116 116 116 116
50 1987 105 105 105 106
55 1904 106 107 107 107
60 1583 101 101 101 101
65 1905 137 137 137 137
70 1524 122 123 123 123
75 1344 117 117 117 117
80 911 98 99 98 99
85 442 58 58 58 59
